Abstract: Although CP violation in the B meson system has been well established by the B factories, there has been no direct observation of time-reversal violation. The decays of entangled neutral B mesons into definite flavor states (B-0 or (B) over bar (0)), and J/psi K-L(0) or c (c) over barK(S)(0) final states (referred to as B+ or B-), allow comparisons between the probabilities of four pairs of T-conjugated transitions, for example, (B) over bar (0) -> B- and B- -> (B) over bar (0), as a function of the time difference between the two B decays. Using 468 X 10(6) B (B) over bar pairs produced in Y(4S) decays collected by the BABAR detector at SLAC, we measure T-violating parameters in the time evolution of neutral B mesons, yielding Delta S-T(+) = -137 +/- 0.14(stat) +/- 0.06(syst) and Delta S-T(-) = 1.17 +/- 0.18(stat) +/- 0.11(syst). These nonzero results represent the first direct observation of T violation through the exchange of initial and final states in transitions that can only be connected by a T-symmetry transformation.